Classical N, P, K Factorial Experiment

A classical N, P, K (nitrogen, phosphate, potassium) factorial experiment on the growth of peas conducted on 6 blocks. Each half of a fractional factorial design confounding the NPK interaction was used on 3 of the plots.

npk

Format

The npk data frame has 24 rows and 5 columns:

block

which block (label 1 to 6).

N

indicator (0/1) for the application of nitrogen.

P

indicator (0/1) for the application of phosphate.

K

indicator (0/1) for the application of potassium.

yield

Yield of peas, in pounds/plot (the plots were (1/70) acre).

Note

This dataset is also contained in R 3.0.2 and later.

Source

Imperial College, London, M.Sc. exercise sheet.

References

Venables, W. N. and Ripley, B. D. (2002) Modern Applied Statistics with S. Fourth edition. Springer.

Examples

options(contrasts = c("contr.sum", "contr.poly"))
npk.aov <- aov(yield ~ block + N*P*K, npk)
npk.aov
#> Call:
#>    aov(formula = yield ~ block + N * P * K, data = npk)
#> 
#> Terms:
#>                    block        N        P        K      N:P      N:K      P:K
#> Sum of Squares  343.2950 189.2817   8.4017  95.2017  21.2817  33.1350   0.4817
#> Deg. of Freedom        5        1        1        1        1        1        1
#>                 Residuals
#> Sum of Squares   185.2867
#> Deg. of Freedom        12
#> 
#> Residual standard error: 3.929447
#> 1 out of 13 effects not estimable
#> Estimated effects may be unbalanced
summary(npk.aov)
#>             Df Sum Sq Mean Sq F value  Pr(>F)   
#> block        5  343.3   68.66   4.447 0.01594 * 
#> N            1  189.3  189.28  12.259 0.00437 **
#> P            1    8.4    8.40   0.544 0.47490   
#> K            1   95.2   95.20   6.166 0.02880 * 
#> N:P          1   21.3   21.28   1.378 0.26317   
#> N:K          1   33.1   33.14   2.146 0.16865   
#> P:K          1    0.5    0.48   0.031 0.86275   
#> Residuals   12  185.3   15.44                   
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
alias(npk.aov)
#> Model :
#> yield ~ block + N * P * K
#> 
#> Complete :
#>          (Intercept) block1 block2 block3 block4 block5 N1 P1 K1 N1:P1 N1:K1
#> N1:P1:K1  0           1     -1     -1     -1      1      0  0  0  0     0   
#>          P1:K1
#> N1:P1:K1  0   
#> 
coef(npk.aov)
#> (Intercept)      block1      block2      block3      block4      block5 
#>  54.8750000  -0.8500000   2.5750000   5.9000000  -4.7500000  -4.3500000 
#>          N1          P1          K1       N1:P1       N1:K1       P1:K1 
#>  -2.8083333   0.5916667   1.9916667  -0.9416667  -1.1750000   0.1416667 
options(contrasts = c("contr.treatment", "contr.poly"))
npk.aov1 <- aov(yield ~ block + N + K, data = npk)
summary.lm(npk.aov1)
#> 
#> Call:
#> aov(formula = yield ~ block + N + K, data = npk)
#> 
#> Residuals:
#>     Min      1Q  Median      3Q     Max 
#> -6.4083 -2.1438  0.2042  2.3292  7.0750 
#> 
#> Coefficients:
#>             Estimate Std. Error t value Pr(>|t|)    
#> (Intercept)   53.208      2.276  23.381  8.5e-14 ***
#> block2         3.425      2.787   1.229  0.23690    
#> block3         6.750      2.787   2.422  0.02769 *  
#> block4        -3.900      2.787  -1.399  0.18082    
#> block5        -3.500      2.787  -1.256  0.22723    
#> block6         2.325      2.787   0.834  0.41646    
#> N1             5.617      1.609   3.490  0.00302 ** 
#> K1            -3.983      1.609  -2.475  0.02487 *  
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Residual standard error: 3.942 on 16 degrees of freedom
#> Multiple R-squared:  0.7163,	Adjusted R-squared:  0.5922 
#> F-statistic: 5.772 on 7 and 16 DF,  p-value: 0.001805
#> 
se.contrast(npk.aov1, list(N=="0", N=="1"), data = npk)
#> [1] 1.609175
model.tables(npk.aov1, type = "means", se = TRUE)
#> Tables of means
#> Grand mean
#>        
#> 54.875 
#> 
#>  block 
#> block
#>     1     2     3     4     5     6 
#> 54.03 57.45 60.77 50.12 50.52 56.35 
#> 
#>  N 
#> N
#>     0     1 
#> 52.07 57.68 
#> 
#>  K 
#> K
#>     0     1 
#> 56.87 52.88 
#> 
#> Standard errors for differences of means
#>         block     N     K
#>         2.787 1.609 1.609
#> replic.     4    12    12